#isotropic

[tenth-sentence]

He followed the same intellectual path, however, assuming an isotropic and homogenous distribution of matter in the cosmos, and searching for solutions to Einstein's equations that describe the story of this smooth and uniform universe.

"Human Universe" - Professor Brian Cox and Andrew Cohen

[waybackwanderer]

The Isotropic Media Group Oct 1996 Archived Web Page

[katieprattartist]

My titles have become more esoteric over time… this one is comparatively self-explanatory 1. Isoberg 2003 #oiloncanvas 200 x 180cm There are skeins of #lemonyellow dotted lines joining similarities across the #pictureplane Voluminous lumps of #paint as well as pigment stencils of splashes lie on the raw #canvas Raw #pigment and fat #oilpaint 2. Cover of #anmagazine March 2004 3. #detail of Isoberg 9 #oilpainting #contemporaryart #contemporarypainting #abstractpainting #abstraction #isotropic #artistsarchive #archive #painting #painter #artist #artistsoninstagram #londonartist #katiepratt #explorepage #artistsnewsletter @anartistsinfo 📸 #RichardKearns https://www.instagram.com/p/CoUsY1godU5/?igshid=NGJjMDIxMWI=

[conceptofjoy]

davesprite looking between john and his corpse on his quest bed. u gonna do anything with that orrrr.

[kilobran]

Love it when my assignment for class is "draw a character" I was gonna do that anyways, so eat that

[hsmagnet]

Mitre Magnet - A Revolutionary Advancement In Precision Magnet

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[tanadrin]

And God said, "Behold! I have created the fourth primordial force: the weak interaction!"

And the angels all clapped and nodded politely, and there was a long silence; and finally Verchiel, the Angel of Grace, spoke up and asked, "Er, what exactly does it do, O Fashioner?"

And God said, "What do you mean, 'what does it do?' It's the fourth fundamental force of the universe."

And Verchiel said, "You mentioned that. Um. But it's just that the other three sort of have a brand, you know? Gravity helps build large-scale structures, acts over vast cosmic distances, shapes time and space. The strong force is secret, hidden, binding together quarks and all that. Electromagnetism, very cool stuff, somewhere in between. We're all big fans of the whole magnetic monopole double bluff, very clever. But, er. What does this 'weak interaction' do?"

And God said, "It mediates radioactive decay. Sort of."

And Verchiel said, "Radioactive decay? All radioactive decay?"

And God said, "No. Just some kinds."

And Zephaniel, the Chief of the Ishim spoke, and he said, "A whole independent force just to mediate some kinds of radioactive decay?"

And God said, "Well. Not totally independent. Technically it's related to electromagnetism."

And Zephaniel said, "Wait, it's not even a real force?"

And God said, "It's totally a real force. It's just that it's one aspect of a combined electromagnetic and weak force. An electro-weak force, if you will."

And Metatron, the Celestial Scribe, scratched his head at this, but said nothing.

And Cambiel, the Angel of Transformation, said, "Maybe you can walk us through it from the top."

And God Sighed an immense Sigh, and said, "All right, fine.

"So the way it works is that all of space and time is permeated by a field that has imaginary mass."

And Cambiel said, "Imaginary mass, O Generous Provider?"

And God said, "Yes, imaginary mass. It's tachyonic, d'you see?"

And Sarathiel, the Angel of Discipline, said, "Wait a minute, I thought we agreed nothing was going to travel faster than light? All that 'c' business and the whole Lorentz transformation thing. What's happening with that?"

And God said, "Let me finish. The field is tachyonic. The particles in the field all move slower than light."

And Sarathiel had to think about this for a second.

And God said, "The point is, a field with imaginary mass has a non-zero vacuum expectation value."

And this really gave Sarathiel trouble, since he had never been very good at math.

And God, seeing this, went back to explain. "Most fields, like the electromagnetic field, have no effect when they are at their lowest energy state. It's like they're not there at all. If you give a field imaginary mass, then it vanishes only when it's at a very high energy state, and at a low energy state, it has a nonzero value everywhere."

And Sarathiel nodded, but he was confused, because he didn't understand why God would create such a thing.

But Verchiel thought he saw where God was going with this, and he was amazed.

"Truly, you are cunning beyond measure, O Only One Certainly Sound and Genuine in Truth! Only now do I understand your design! For in order to make the universe homogenous and isotropic, it is necessary that all large-scale fluctuations in temperature and mass must be evened out early in the history of the cosmos; and therefore, you have designed a field which will rapidly expand space after the Big Bang, many orders of magnitude in brief moments, and then swiftly and spontaneously decay as it gives up the energy it began with, giving rise to radiation and particles of all kinds as it does, which will condense into the material universe! It is a wonder to behold."

And God said, "What? No. I mean I did, but this isn't the inflaton field I'm talking about. This is something else."

And Verchiel said, "Wait, it's not?"

And God said, "No, I'm going to use a different field to drive cosmic inflation. The properties of this field are totally different."

And now Verchiel was also confused, and lapsed into silence.

And God said, "Like I was saying, this field is a scalar field with imaginary mass, and it does spontaneously decay to a ground state with a non-zero value. But it's not the inflaton field. Instead it combines with the W1, W2, W3, and B bosons."

And Metatron began to flip back through the pages of the Heavenly Record trying to figure out where he'd lost the thread.

And Zephaniel said, "The what bosons?"

And God said, "The W1, W2, W3, and B bosons. I'm sure I mentioned them. You know, the massless bosons?"

And Zephaniel said, "I'm pretty sure we only talked about the W+, W-, and Z0 bosons. All of which you said were going to have mass, O Owner of All Sovereignty."

And God said, "Yes, but this is how they get them, you see. Once this field acquires a nonzero value everywhere, the massless bosons interact with it and get mass. Well, some of them do. They turn into the W+, W-, and Z0 boson. And the photon."

And Zephaniel said, "…and the photon, O Accepter of Invocation?"

And God said, "Well, I did say I was going to unify the electromagnetic force and the weak interaction, didn't I? This is how. Above the critical temperature--right now I'm thinking 10^15 K, but I'm open to feedback on that one--electromagnetism and the weak force act as a single unifying force. Below that temperature, the field gets a nonzero value, you get three massive bosons to mediate the weak interaction, and the photon pops out seperately."

And Zephaniel said, "That seems… a bit overly complicated, doesn't it, O Reinstater Who Brings Back All?"

And God said, "No, it's exactly what we need. Look, that way the W and Z bosons have something to do, but the weak interaction still only travels short distances. Gravity is still the star of the show on cosmic scales, as it were. But now quarks and leptons can swap their flavor!"

And Zephaniel said, rather weakly, "Their… flavor, O Source of Good?"

And God said, "It's this new quantum number I'm trying out, to give the three generations of matter more unique identities."

And Cambiel said, "Three generations of matter? Now I'm really confused."

And God said, "I'm sure I mentioned this. You've got the lightest quarks and leptons, and then two heavier versions of each that can decay into the lighter versions."

And Cambiel said, "What do they do? New kinds of chemistry, is it?"

And God said, "Well, no. Mostly they just decay in a couple microseconds. Or even faster."

And Zephaniel began to rub his temples, and Cambiel sniffed.

And Cambiel said, "This all seems a bit ad hoc to me. Not really the stuff of an elegant and obviously ordered Creation. Why not have four generations of matter? Why not a trillion?"

And God began to grow irritable, and said, "Well, that's not really up to you, now is it? We're going to have three generations of matter, and the electroweak force, and that's that!"

And Zephaniel said, "As long as we are unifying fundamental forces, perhaps we could somehow also unify the electroweak interaction with the strong interaction, or even gravity."

And God hesitated saying, "Well, I haven't decided about that yet. I'm not sure I want gravity to be quantized, you know? Seems to take some of the geometric elegance out of general relativity."

And now it was Zephaniel's turn to sigh, and he bowed his head. "As you wish, O Possessor of Authority of Decisions and Judgement."

[mpcomagnetics]

Where are Samarium Cobalt Magnets Useful?

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[just-here-with-my-thoughts]

It's Zircon Time!

(If you are the same age as me you may well have read that like you were a Power Ranger saying "It's Morphin' Time" and you wouldn't be incorrect because we're going to talk about how zircon slowly degrades from crystalline to amorphous structure over millions of years... so...)

I'm excited to tell you about gem-quality zircon!

@theproblemwithstardust I have no idea if you're still interested but heads up I got carried away I think I wrote over 3k words about this you may need snacks and an interval

Zircon & Double Refraction

Zircon is a naturally occurring gem mineral, chemical formula zirconium silicate and crystallising in the tetragonal system. It is a uniaxial optically anisotropic gemstone (remember, optically anisotropic means light is split into two when passing through the crystal) and in its rough form occurs as elongate to squad tetrahedral crystal with bipyramidal terminations.

(What does that mean? It means it's a rectangle with two triangles at the ends, and the rectangular section might be long or short!)

Even as a rough crystal zircon will show some of the features it is well known for - such as its sub-adamantine lustre (meaning the near-diamond-like brightness of surface reflection of white light) and (if the crystal is transparent enough) a high birefringence value means internal features are viewed as doubled.

Woah... you just chucked a LOAD of science terms at me! What does any of it mean?

Let me tell you about double refraction in zircon! I'm so excited by it!

I'll break these terms down one at a time:-

Refraction - the bending of light as it passes through the crystal

Double refraction - two rays of light get bent at slightly different angles! (Remember that as an optically anisotropic material, zircon splits light into two rays)

Birefringence - the difference in the amount the two rays of light are bent by

You know how when you stick a straw in a glass of water, it looks as though it is slightly bent? That's refraction at play! When light passes between two mediums of different optical densities (for example from air into water, or air into a gemstone) the light is bent. The angle is it bent by is related to the substance it passes through. We use refraction in gemmology to identify gemstones, because every stone refracts (bends) light by a different amount.

In an optically anisotropic gemstone the two rays of light are both bent by different amounts. We can measure how much each ray of light is bent by, and the difference between them, to help identify them!

Fun fact, the 2 rays of light are referred to as the ordinary ray and the extraordinary ray - that is important for identifying stones by their RI but I'm getting off topic for zircon!

The important thing in zircon is that it has Very High Birefringence. That means that the two rays of light are bent by so much that when they leave the gemstone and reach the viewer (that's you!) you see a doubled image of whatever is inside the stone - double vision!

(If like me you wear glasses, it feels like looking at the gemstone without your glasses on - everything just sliiightly out of focus...)

If a zircon has inclusions, each of these will appear to be doubled when viewed through the crystal. Even easier to spot, and present even in an inclusion-free stone, is the doubling of the back facets of the gemstone! That's right, when you look through the stone at the pattern of facets on the other side, they will appear to be doubled.

Haang on a sec - what about the Optic Axis, I hear you ask?

An Optic Axis is a direction in an optically anisotropic gemstone in which light behaves as though it is passing through an optically isotropic material. That's a material where light travels as a single ray rather than splitting into two - so when viewed from the right angle, a zircon crystal will let light pass through as a single ray, and you won't see any double refraction at all!

Fun fact! In a uniaxial crystal, there is one optic axis and it is always parallel to the c-axis (the long dimension of the crystal).

This is all pretty neat, right? If you want your colourless zircon crystal to pass as a diamond imitation, it should be cut with the table perpendicular to the c-axis. That way, when you look straight down at the top of the stone, you won't see any of that dizzying eye-visible double-refraction - diamonds are optically isotropic, so they only ever transmit a single ray of light :)

(I mean there are tons of other ways to differentiate diamond and zircon, but at a glance, it would make for a better imitation...)

Metamict Zircon

Wouldn't it be great if zircon were always so easily identifiable in part due to its high birefringence? Not a lot of stones that have eye or loupe visible birefringence - most of them are much smaller values (ie. the difference in the angle the two rays of light are bent at is much smaller)

Sad news for you friends, but zircon does not always stay in this nicely ordered highly crystalline state, behaving as a tetragonal optically anisotropic crystal should.

You see, the thing that gives zircon it's colour is Uranium. That's right, radioactive uranium!

And even when it is only present as a few ppm (parts per million) in the zircon structure, the radiation emitted by those atoms is enough to start breaking through the bonds between other atoms in the zirconium silicate structure, and slowly but surely the structure of zircon is transformed from highly organised crystalline bonds, to irregular and disorganised amorphous atomic arrangement.

Why am I talking about this off the back of birefringence values?

Because amorphous materials are optically isotropic - that is, light behaves the same in all directions. No more double refraction!

(In case you haven't noticed yet you are actually taking a stroll through the gemmology section of my mind and encountering Related Thoughts in the exact order in which they are stacked and catalogued in my Mental Shelves, I hope you are keeping up but hit me up in the comments or reblogs if you need me to circle back to anything?)

But! I hear you cry - if you identify zircon by it's birefringence, how do you tell it's zircon when it's altering to an amorphous state?

When zircon becomes amorphous - also known as becoming metamict, it doesn't suddenly change all at once. There will be areas of crystalline structure interspersed with amorphous areas. When viewed with a 10x loupe, this gives the stone a hazy or grainy internal appearance. Although many gemstones show various types of zoning, the hazy zoning associated with metamict zircon is quite distinctive.

The colour of zircon also changes during this breakdown of the crystal structure, becoming a greenish colour instead of the usual browns, reds, yellows.

Heated zircon can be colourless or even bright blue - I kinda assume they would follow the same path when becoming metamict but can't say for sure! That said, I know from my own experimentation that heated blue zircon goes a murky grey-brown on exposure to UV, so perhaps it really is all the same process :)

Diagnostic Absorption Spectra in Zircon

What if the zircon is completely metamict? What if there is no birefringence at all, and you can't be certain that the haziness inside the stone is identifiable as hazy zoning?

Well, don't forget that a lot of gemmological identification involves combining different observations and test results - it's very rare to be able to do a single test and say "THIS. It's definitely THIS."

That said... there is a test we can do on zircon which provides a diagnostic result, even in the absence of all other tests!

We can check the Absorption Spectrum!

Holllllld up... better go over what we mean by Absorption Spectra.

When white light passes through a gemstone, it is modified via absorption...

Wait, go back a bit further.

White light is actually made up of light of all coloured wavelengths from 400nm (violet) to 700nm (red) - this is called the 'visible light spectrum' as these are the wavelengths the human eye can detect!

When all colours of light at once reach the human eye, the mind interprets it as 'white' light.

When the colour is modified because one or more wavelengths are removed, we then start to perceive colour.

For example, when light passes into a corundum crystal coloured by chromium, most violet, some blue, and all green and yellow and some orange light is absorbed. The resulting colour you see, made up from transmitted red and a little blue light, is red!

By the way, the absorption I just described is for ruby! :)

BACK TO ZIRCON!

The uranium in zircon causes a unique absorption which can be viewed with a spectroscope. Many gemstones, even colourless ones, show various absorption spectra. Relatively few show a diagnostic absorption spectra, which has a pattern of absorption unique to that gem species and colouring element. Although you should always back up your gemstone identifications with multiple pieces of evidence, with a diagnostic feature you can make a positive identification of gem species even in the absence of other information.

So what does zircon's diagnostic absorption spectrum look like?

Well, that can vary depending on whether it is high (crystalline) zircon or low (metamict) zircon, and whether it has been heat-treated to alter it's colour!

The defining feature in all types of zircon is a diagnostic absorption line at 653nm (about half-way along the red area of the spectrum).

In heat-treated colourless or blue zircon, this may be the only absorption line present.

In other zircons, as well as the 653nm line, you will see up to 40 other absorption lines or bands scattered throughout the spectrum.

(Oh backtrack a moment again! When viewing the absorption spectra, the black lines are the wavelengths that are being absorbed :) Please do remind me to tell you this stuff, I forget because I Thought It Was Obvious (as Tech would say) but I'm doing my best to cover the basic science as well as gemmology here)

Metamict zircon still shows a diagnostic absorption spectra, including the line at 653nm. However, many of the absorption lines and bands have become distinctly fuzzy-looking - this is another helpful piece of evidence to determine whether you are looking at high zircon or metamict zircon.

(Still here, team? I have been writing for 2 hours and we're 1700 words deep, so take a break if you need it, hydrate or diedrate... save this post for later... close the tab and back slowly away from the screen because you didn't realise you were getting into all this when you clicked below the read-more...)

Toughness and Dispersion

Other features of zircon that are important when considering its use as a gemstone are it's relatively high hardness, but low toughness.

Hardness and toughness? Aren't they the same thing?

No :D

Hardness (in gemmological terms) relates to the ability of a material to resist being scratched when the sharp point of another material is dragged across the surface. Zircon has a Moh's Relative Hardness rating of 8/10 - that's pretty good resistance to being scratched!

However, it has Low toughness. Toughness relates to the ability of a material to resist being fractured or cleaved as a result of physical impact. In short, zircon chips easily. When faceted as a gemstone, it tends to chip along the sharp edges between the facets, accumulating numerous small fractures which are described cumulatively as 'nibbled facet edges'.

Remember when I said there were numerous other ways to tell the difference between diamond and zircon? This is one of them! You would not expect to see wear like this to a diamond - however, unless your zircon has been very well cared for, you would expect the facet edges to show a certain amount of abrasion due to its low toughness... and yes, if you view those chips through the crystal, they will appear to be doubled due to the high birefringence :)

Other features of zircon include a high Dispersion Index.

Dispersion is the splitting of white light into its spectral wavelengths when passing through two inclined surfaces of a transparent material (think the Pink Floyd Dark Side of the Moon prism splitting the single ray of white light into the rainbow!)

Guess what? When you facet a gemstone, it almost always has two inclined surfaces of transparent material! In gemstones with a high Dispersion Index, you will see more dispersion - that is, when you tilt them, you will see more flashes of coloured light (blue, green, red) sparkling back at you from the facets than a gemstone with a low dispersion index.

You can see dispersion even in gemstones with a body colour - they don't have to be colourless to see it! That said, it can be harder to spot some of those dispersed colours against the body colour - so this feature may be more prominent in colourless or light coloured zircons than in deeper colour samples.

Aaaaaaaand I think that's it? At least, I think that's everything I wanted to cover with you about gem-quality zircon today!

BUT WAIT!

The fun's not over :P

I'm pretty sure I promised a comparison between naturally occurring zircon, and common artificial material cubic zirconia!

(Just when you thought you were free...)

Natural vs. Artificial

Zircon is a naturally occurring, crystalline, inorganic mineral formed by natural processes. Don't forget, the chemical formula is Zirconium Silicate and it crystallises in the tetragonal system.

Cubic Zirconia (shortened to CZ) is an artificial crystalline material grown by man. The chemical formula is Zirconium Oxide and it crystallises in the cubic system.

Notice how I say artificial, not synthetic? That's an important distinction.

Synthetic materials have a direct comparison in nature. All synthetic gemstones are artificially grown, but not all artificial materials are considered synthetic.

To be classified as a synthetic gemstone, the result of the artificial growth process must be chemically, physically and optically identical to the naturally occurring mineral.

Cubic Zirconia has no natural counterpart. It is completely artificially created. Hence, it can be referred to as artificial or as a simulant (meaning it is simulating/imitating another natural material).

Cubic Zirconia

CZ is grown via the incredibly metal sounding process of SKULL MELTING

Fun fact! CZ has a higher melting point than any material you could make a crucible of to melt it in. So when we make it, we have to melt it inside a skin, or 'skull', of solid CZ, with a liquid inside. We can go into the science another time if you like, but the quick version is it's like heating up lasagne in the microwave, and the middle gets hot whilst the edges are still cold... that's how we melt the centre of the CZ mix whilst keeping in in a cooled skull of its own solid material :)

Above a certain temperature (I want to say off the top of my head with a melting point in excess of 2600 degrees Celsius), zirconium oxide adopts a cubic arrangement - nice and symmetrical, and optically isotropic (light moves as a single ray and behaves the same in all directions). However below temperatures of around 2000 degrees Celsius, zirconium oxide would crystallise in the monoclinic system. Woaaah! That's not terribly symmetrical! We want it to be cubic zirconia, the same in all directions.

To stop the zirconium oxide mix from changing to a monoclinic arrangement as it cools, we need to introduce a stabilising element. Most often this is a rare earth element such as yttrium, which bonds with the zirconium oxide structure and forces it to retain its cubic arrangement even at lower temperatures. Cool, right?

(ahaha I didn't mean to make that joke... cool... cos we're cooling the mixture... it's late and I've been typing for a long time now I hope you're still with me...)

Fun fact, due to it's super high melting point, you actually need to start the melting and recrystallisation process by inserting a thin wafer of pure zirconium into the ingredient mix inside the Skull. This can then be melted, which will then oxidise, and start a chain reaction of melting and bonding with the surrounding material :)

CZ Optical Behaviours

CZ is cubic, meaning it is optically isotropic, so you will only have single refraction (a single ray of light transmitted) no matter which direction you view the stone in.

This is enough to differentiate it from zircon, which is optically anisotropic with high birefringence, but what about diamond, the gemstone that CZ so frequently imitates?

Don't worry - we can again look at features such as the absorption spectra, and hardness/toughness as well as lustre to tell the difference.

The lustre (surface reflection of white light) of CZ is lower than that of diamond (adamantine) or zircon (sub-adamantine). I want to say CZ is bright vitreous (bright glasslike)? So with practice you will learn when the surface reflection just doesn't seem quite bright enough to be diamond...

(It's bothering me that I skipped the formal lustre definition earlier so here it is now: lustre is the quality and quantity of white light returned via reflection from the surface of the gem material towards the viewer)

CZ has a really high dispersion index - that's the splitting of white light into its spectral colours, remember, the coloured sparkles you see when you move the stone! CZ has a dispersion index even higher than diamond, so if the stone seems to be returning too many sparkles of colour... it's probably too good to be true.

CZ also grows free of inclusions, and the absence of internal features is a huge warning sign that you're looking at an imitation rather than a diamond!

As well as these optical features, CZ will also show surface features to differentiate it from zircon or diamond. The facet edges even in well-cut stones are typically rounded and soft, and although they don't tend to accumulate the 'nibbled' look of chipped zircon, the facets can be scratched - typically picking up a 'frosted' look if they are particularly heavily worn.

Remember when I mentioned using yttrium or a similar element to force the zirconium oxide mix to retain it's cubic structure? That's what causes the absorption spectra in CZ!

CZ shows a REE (rare earth element) spectra. It's not diagnostic like a zircon spectra is, but it is highly characteristic. Both coloured CZ and colourless can show it - REE spectra typically manifest as clusters of many fine absorption lines mostly distributed in the yellow-green area of the spectrum.

Iiiiiii think I might actually be done now (except the zircon vs CZ comparison strayed into grounds of differences between diamond and its common simulants so I'm trying not to get doubly triply side-tracked into synthetic moissanite which is ALSO doubly refractive with loupe-visible doubling of internal features due to high birefringence and only gets to old the term 'synthetic' on a technicality because naturally it is almost impossible to find gem-quality moissanite outside of rare meteoric impact events...)

(Guess what synthetic moissanite is silicon carbide crystallising in the hexagonal system and is created by a process known as sublimation - where a material goes from solid to gaseous or plasma state (or back) without passing through a liquid state in between!)

Hmm I realise by the time I post this I will have added photos but I think I'll save as a draft for now and maybe sleep before I do that... **Photos added now! Microscope still not set up so you have to make do with my internet searching, sorry

I hope you have enjoyed today's citizen science gemstone lecture brought to you by the ability to recall memorised information without checking my notes and the parasocial relationship I have projected onto you, the Tumblrites, hoping that you will love the science side of gemstones as much as I do!

just-thoughts-about-gems (this is the place to ask your gemstone questions)

[hsmagnet]

High End Magnets for Particle Accelerators

High End Magnets for Particle Accelerators High End magnets are often used in very sensitive measurement and laboratory equipment. The major difference between standard magnets and high end magnet is the high level of narrow specifications and documentations made on each magnet. In order to be able to deliver high end magnets HSMAG have build up strong competencies within magnetic measurements…

[fumblebeefae]

If you’re looking at this, you are one of the first in the world to see the brain cells of the Australian stingless bee (Tetragonula carbonaria).

I’ve been learning the isotropic fractionator method to count brain cells in the brain. The plan is to apply this method to Australian bee species to find out the number of neurons in their brains.

[willigula]

Isometric Systems in Isotropic Space. Map Projection: Snail, by Ágnes Dénes, 1974

[sukimas]

give me 2 million dollars

i made the empire state building with magnets and radio waves

[mpcomagnetics]

The Complete Guide of Neodymium Magnets

The Complete Guide of Neodymium Magnets NEODYMIUM MAGNETS: THE STRONGEST COMMERCIALLY AVAILABLE MAGNETS Neodymium Magnets are really Strong Magnets made by an alloy of neodymium, iron and boron (NdFeB). Neodymium is a rare-earth element (REE), also called rare-earth metal or (in context) rare-earth oxide. Even at their smallest size Neodymium Magnets have a Really High Magnetic Strength. They…

[rock-swag-tournament]

Mineral Swag Round 3: Gems

Aquamarine is the blue variety of beryl, which is a cyclosilicate mineral with beryllium in it. Aquamarine is named for sea water - literally aqua marina aka "marine water." It is a beautiful gemstone and the March birthstone.

Garnet is the January birthstone and actually comes in MANY colors. It is commonly red or pink but can be green or orange as well, depending on the garnet variety you’re talking about. It naturally forms dodecahedra and is an isotropic mineral so it goes “extinct” (turns opaque) under cross polarized light. Pictures of this are in previous rounds!

Garnet is, by far, my favorite mineral because of almandine garnets (shown) are such a beautiful deep red color. Plus they're a metamorphic mineral! So they form in rocks like schist and gneiss. Aquamarine, for anyone interested, forms in igneous rocks like granite.

[tensorboi]

Why is the Pythagorean theorem true, really? (and a digression on p-adic vector spaces)

ok so if you've ever taken a math class in high school, you've probably seen the Pythagorean theorem at least a few times. It's a pretty useful formula, pretty much essential for calculating lengths of any kind. You may have even seen a proof of it, something to do with moving around triangles or something idk. If that's as far as you've gotten then you are probably unbothered by it.

Then, if you take a math class in university, you'll probably see the notion of an abstract vector space: it's a place where you can move things and scale them. We essentially use these spaces as models for the physical space we live in. A pretty important thing you can't do yet, though, is rotate things or say how long they are! We need to put more structure on our vector spaces to do that, called a norm.

Here's the problem, though: there are a *lot* of different choices of norm you can put on your vector space! You could use one which makes Pythagoras' theorem true; but you could also use one which makes a³ + b³ = c³ instead, or a whole host of other things! So all of a sudden, the legitimacy of the most well-known theorem is called into question: is it really true, or did we just choose for it to be true?

And if you were expecting me to say "then you learn the answer in grad school" or something, I am so sorry: almost nobody brings it up! So personally, I felt like I was going insane until very recently.

(Technical details: the few that do bring it up might say that the Pythagorean norm is induced from another thing called an inner product, so it's special in that way. But also, that doesn't really get us anywhere: you can get a norm where a⁴ + b⁴ = c⁴ if you are allowed to take products of 4 vectors instead!)

How is this resolved, then? It turns out the different norms are not created equal, and the Pythagorean norm has a very special property the others lack: it looks the same in every direction, and lengths don't change when you rotate them. (A mathematician would say that it is isotropic.) Now, all of a sudden, things start to make sense! We *could* choose any norm we like to model our own universe, but why are we going to choose one which has preferred directions? In the real world, there isn't anything special about up or down or left or right. So the Pythagorean norm isn't some cosmic law of the universe, nor is it some random decision we made at the beginning of time; it's just the most natural choice.

But! That's not even the best part! If you've gone even further in your mathematical education, you'll know about something called p-adic numbers. All of our vector spaces so far have been over the field of real numbers, but the p-adic numbers can make vector fields just as well. So... are the Pythagorean norms also isotropic in p-adic spaces? Perhaps surprisingly, the answer is no! It turns out that the isotropic norms in p-adic linear algebra are the ∞-norms, where you take the maximum coordinate (rather than summing squares)! So the Pythagorean theorem looks very different in p-adic spaces; instead of a² + b² = c², it looks more like a^∞ + b^∞ = c^∞.

If you're burning to know more details on this, like I am right now as I'm learning it, this link and pregunton's linked questions go into more details about this correspondence: https://math.stackexchange.com/questions/4935985/nature-of-the-euclidean-norm

The interesting thing is that these questions don't have well-known answers, so there is probably even more detail that we have yet to explore!

tl;dr: the pythagorean theorem is kind of a fact of the universe, but not really, but it kinda makes sense for it to be true anyway. also we change the squares to powers of infinity in p-adic numbers and nobody really knows why